Is real analysis really important in the process of learning statistics?

The reason is, I suck at real analysis(actually, I failed it) and do really good at stats(I am in my third yr. at uni.).

Should I continue studying stats or switch to another major?

I am looking for some advice, plz.

Hey Stat313 and welcome to the forums.

For your question my answer is yes and no depending on what kind of statistics/probability you are doing.

For many purposes I would say that it is not absolutely essential to have a solid understanding in real analysis.

It depends on what kind of work you're doing and what kind of problems you are working on. If you are dealing with say the theoretical side of stochastic calculus and you have to understand for example when a particular stochastic process makes sense or how to deal with it, then yes real analysis is absolutely vital.

Also in terms of stochastic processes in general, if you want to rigorously work with something that is well defined you have to resort to using measure theory of which the measure you are using is probabilistic in nature (I think they call them Borel measures if I remember correctly) and this means you go through the whole measure theory blah blah blah to analyze it in this context.

Now if you primarily want to use others results that are derived from theoretical statisticians or pure mathematicians but still have to do something analytic where you are actually doing the 'statistical' work without worrying about the theoretical foundations (i.e. you let the theoretical guys check it out) then this should be ok and in fact many statisticians are, in my guess, actually doing this anyway.

The reason I say this is, is because the same kind of thing happens in engineering and even in applied mathematics in some contexts. Basically as long as the methods are sound and as long as you understand the assumptions and what they really mean (very important point here), then there is no reason why you need to prove everything every time or even know the nuances. But what it does mean is that if you can't use a particular method because of assumptional circumstances and you need to use another method, then you will need to make sure that is sound either by proving it yourself or getting someone else to do it.

Again it depends on what kind of systems you are working on. If you are dealing with systems of infinite random variables, you definitely will need to know real analysis and probably functional analysis as well.

If you are more concerned with things involving designing and analyzing experiments or doing computational analysis where the procedures are proven to work and the assumptions clearly understood, then I don't think real analysis is going to be required.

Personally I think you'll be able to find lots of work where you don't need a lot of real analysis and where you can still do some deep statistical work.

Just be aware of the kinds of things that will require a deep knowledge of real analysis: these kinds of things include stochastic calculus where you have a system that has not been strictly dealt with and it's properties investigated and proofs formed but where you still need to know things like say if the process is continuous, if it converges and also things to do with calculus like integration.

I would talk to other people in math at your school. In a lot of places (but not all), analysis is kind of like a hoop to jump through so they can weed out weaker students. It's not necessarily linked to other courses you will take. But it's going to depend a lot on what your program is like. In the program I took, some concepts from analysis came up again but analysis-style thinking didn't really come back.

In general? No. Not for most applied purposes.
Keep in mind that rigorous probability theory requires measure theory, which will require you to pick up some real analysis. The frontiers of statistics nowadays use everything from functional analysis to differential and algebraic geometry.